The comments on the question point out that it's not really well-posed: the property "bijective" isn't defined for morphisms of an arbitrary category.
However, for maps between sets, "bijective" means "injective and surjective". A common way to interpret "injective" in an arbitrary category is "monic", and a common way to interpret "surjective" in an arbitrary category is "epic". So we might interpret "bijective" as "monic and epic".
Then JHS's question becomes: is there a name for categories in which every morphism that is both monic and epic is an isomorphism? And the answer is yes: balanced.
It's not a particularly inspired choice of name, nor does it seem to be a particularly important concept. But the terminology is quite old and well-established, in its own small way.
Incidentally, you don't have to interpret "injective" and "surjective" in the ways suggested. You could, for instance, interpret "surjective" as "regular epic", and indeed that's often a sensible thing to do. But then the question becomes trivial, since any morphism that's both monic and regular epic is automatically an isomorphism.
I'm not sure what "topograph" means, but if it's just a fancy word for "pictoral description of features in a region" then the answer is yes.
Send your triple $(a,b,c)$ to the element $a/c+ ib/c$ in the unit circle, and apply the Cayley transform $z \mapsto -i\frac{z-1}{z+1}$ to get elements of the rational projective line. The basic triple $(1,0,1)$ is sent to $0$, and the standard symmetry-breaking restriction specifying that $b$ is even identifies the set of Pythagorean triples with the $\Gamma(2)$ orbit of $0$, i.e., the set of rationals with even numerator. We will call rationals with even numerator "special cusps".
Divide the upper half plane into fundamental domains for $\Gamma(2)$ - each domain will touch a single special cusp, and three non-special cusps. The topograph is formed by dividing the half-plane into regions, where each region is the union of all fundamental domains touching a special cusp. You can also use the Cayley transform to get a topograph in the disc (as in your last picture above), where the special cusps are precisely the rational points on the unit circle whose $y$-coordinate has even numerator.
Unlike the quadratic form topograph, the regions only meet at edges, because the fundamental domain of $\Gamma(2)$ has no corners - this has something to do with the lack of finite order elements. I tried drawing the topograph in a disc, but it doesn't look so great. The regions attached to a single Pythagorean triple are so big that everything except about three regions will be smushed into a small neighborhood of the boundary circle.
Since you added some other information, like $3 \times 3$ matrices and ternary trees, I guess you want to know how they are related to topograph process. The correspondence between the $\Gamma(2)$ action and the $3 \times 3$ matrices is an integral version of the exceptional isomorphism between $PSL_2(\mathbb{R})$ and $SO(2,1)^+$ (but perhaps $PGO(2,1)^+$ is more canonical given the process that follows). For example, conjugation by Cayley sends $U = \binom{12}{01}$ to the Möbius transformation $z \mapsto \frac{(1+i)z + i}{-iz+1-i}$, which takes $a+ib$ to $\frac{-a-2b-2 + (2a+b+2)i}{2a+2b+3}$. We can now view the denominator and the real and imaginary parts of the numerator as row vectors, sending $(a,b,1)^T$ to the given combinations. This yields your second matrix with the first row negated. The sign change doesn't affect the validity of the triples, since it just negates $x$. A similar process with $L = \binom{10}{21}$ yields $z \mapsto \frac{(1-i)z + i}{-iz+1+i}$, taking $a+ib$ to $\frac{-a+2b+2+(-2a+b+2)i}{-2a+2b+3}$, which produces your third matrix. Your first matrix corresponds to $U^{-1}$, and $L^{-1}$ goes to a fourth matrix that you didn't write.
The ternary tree is a quotient of a connected subset of the Cayley graph of $\Gamma(2)/\langle - I\rangle$, which is free of rank 2. Because the subgroup generated by $L$ is the stabilizer of the cusp at 0, the cusps are in correspondence with left cosets, i.e., the set of words in $U$ and $L$ that don't end in a nonzero power of $L$. You can write this coset space as the subset of the Cayley graph (an infinite 4-valent tree) from which two of the four branches incident to the identity have been deleted, i.e., a pair of ternary trees sharing a common root at the identity. The entries in the two ternary trees differ by a sign, so we typically identify entries with the same absolute value. This is described in greater length in the article by Alperin that you linked (the main difference being that Alperin uses an action by conjugation on nilpotent $2 \times 2$ integer matrices instead of Möbius transformations).
Best Answer
There's no more serious name for the topograph, as far as I know. And Conway puts a lot of thought into his names, so I think it's best to keep it. I think it's meant to fit into a larger metaphorical system with his rivers and lakes and climbing, which make it so pleasant to study binary quadratic forms. The "topograph" refers to the topographical map one uses for navigation, I believe.
Here's a description with some words in italics -- searching for these words might help you find better resources. The underlying geometry (forget about quadratic forms for a moment) is the $(\infty,3)$ tiling of the hyperbolic plane, i.e. the one with Schlafli symbol $\{ \infty,3 \}$. This is a tiling of the hyperbolic plane by apeirogons (infinity-gons), meeting three at a vertex. The incidence of these apeirogons, edges, and vertices is the Coxeter geometry associated to the Coxeter group of type $(\infty,3)$.
When drawing topographs on the computer, using the Poincare disk model of the hyperbolic plane, each apeirogon is inscribed in a horocycle. In the disk model, the horocycle appears as a circle. I think it looks good to place the face-labels at the (Euclidean) centers of these circles. One can find these centers easily enough by taking the centers of the Ford circles and mapping them by a Mobius transformation which sends the upper half-plane to the unit disk. I also like to scale the fonts via the hyperbolic metric.
Happy drawing...