Finding all the integral points on an elliptic curve is a non-trivial computational problem. You say you are a "non-professional" so here is a non-professional answer: get hold of some mathematical software that does it for you (e.g. MAGMA), and then let it run until it either finds the answer or runs out of memory. Alternatively, do what perhaps you should have done at the start if you just have one curve and want to know the answer: post the equation of the curve, and hope that someone else does it for you. Here's another example of an algorithm currently used in these sorts of software (a Thue one was mentioned above but here's a different approach): find generators for the group (already computationally a bit expensive at times, depending on your luck and/or the size of sha), invoke Baker-like theorems saying "if the coordinates of the point are integral then it must be of the form sum_i n_i P_i with the n_i at most ten to the billion", and then use clever congruence techniques to massively cut down the search space by giving strong congruences for all the n_i. Then just do a brute force search.
Whether or not this will work for you, I cannot say, because it all depends on how big the coordinates of your curve are. The only clue you give so far is that the conductor is "bigger than 130000" [Edit: that was written before the OP edited the question to tell us which curve he was interested in] which of course does not preclude it being bigger than 10^10^10. Also, you need an expert to decide which of the algorithms is best for you. I'd rather do a massive amount of arithmetic in a field of tiny discriminant than a small amount of arithmetic in a field whose discriminant is so large that I can't even factor it, for example.
So in short the answer is that you're probably not going to be able to do it with pencil and paper, but there are programs around that will do it, if all you want to know is the answer.
EDIT: you posted the equation of the curve. Magma V2.15-10 says the integral points are
[ <-23, -196>, <19, 182>, <61, 784>, <-191, 28>, <103, -1442>, <-19, -144>,
<-67, 592>, <23, 242>, <-49, -454>, <-157, -742>, <817, 21196>, <521, 11364>,
<3857, 200404>, <10687, -910154>, <276251, -118593646> ]
plus what you get if you change all the y's to -y's.
$\newcommand\F{\mathbf{F}}$
$\newcommand\Z{\mathbf{Z}}$
$\newcommand\SL{\mathrm{SL}}$
Because of the existence of the Weil paring,
elliptic curves with such a subgroup only exist when
$p \equiv 1 \mod \ N$.
Let $S_N$ denote the set of elliptic curves over $\F_p$ such
that $E[N]$ is defined over $\F_p$.
It will be slightly easier to assume that $N \ge 3$. In this case,
$Y(N)$ is a fine moduli space, and an $\F_p$-point on $Y(N)$
corresponds to
a pair $(E,\alpha:E[N] \simeq \Z/N\Z \times \Z/N \Z)$ defined over
$\F_p$. Given an elliptic curve $E \in S_N$,
how many points does it contribute to $Y(N)$? For a curve $E$
whose automorphism group is $\Z/2\Z$, We see that out of
the $|\SL_2(\Z/N\Z)|$ possible choices of $\alpha$
(technical remark, we have fixed a Weil pairing so that $Y(N)$ is
connected), $(E,\alpha) \simeq (E,\alpha')$ only if $\alpha' = \alpha$
or $\alpha' = [-1] \alpha$. Thus
$E$ contributes $|\SL_2(\Z/N\Z)|/2$ points to $Y(N)(\F_p)$.
In general, $E$ may have slightly more
automorphisms, and we deduce that (for $N \ge 3$):
$$|Y(N)(\F_p)| = |\SL_2(\Z/N\Z)| \sum_{E \in S_N}
\frac{1}{|\mathrm{Aut}(E)|}.$$
Note that the quantity on the right is very close to
$|\SL_2(\Z/N\Z)| \cdot |S_N|/2$, one only has to worry about the
elliptic curves with $j = 0$ or $j = 1728$, and this can be done by hand
if one wants to cross all the i's and dot all the t's.
Suppose that $X(N)$ has $c_N$ cusps and genus $g_N$ (there are some
explicit slightly unpleasant formulas for these numbers, which can
be found (for example) in Shimura's book. All the cusps
are defined over $\F_p$ (with $p \equiv 1 \mod N$) so by the
Riemann hypothesis for finite fields,
$$|Y(N)(\F_p) - (1+p) + c_N| = |X(N)(\F_p) - (1+p)| \le
2 g_N \sqrt{p}.$$
If $g_N = 0$ (which only happens if $N \le 5$), this leads to an
exact formula for $|S_N|$.
In general, at least for large $p$, we see that
$$|S_N| \sim \frac{2p}{|\SL_2(\Z/N \Z)|}.$$
To make this all completely explicit for $N = 3$ (for example),
one gets, presuming I have not made a horrible computational error which is quite possible:
$$S_3 = \begin{cases} (p+11)/12, & p \equiv 1 \mod \ 12, \\\
(p+5)/12, & p \equiv 7 \mod \ 12. \end{cases}$$
(note that $p \equiv 1 \mod 3$):
Of course, "exact formulas" will only exist for $N \le 5$.
Some related and slightly more difficult counting problems are also
nicely explained by Lenstra here (See 1.10):
https://openaccess.leidenuniv.nl/bitstream/1887/3826/1/346_086.pdf
Best Answer
This question is discussed very carefully in Section 3 of the paper Mod $p$ representations on elliptic curves, by Frank Calegari (available here).
In particular, after noting that the answer is positive when $p \leq 5$ (as was already observed in the comments above), he proves (in Theorems 3.3 and 3.4) that if $p \geq 7$ then there exists a continuous representation $\rho: Gal(\overline{\mathbb Q}/ \mathbb Q) \to GL_2(\mathbb F_p)$ with cyclotomic determinant which does not come from the $p$-torsion of an elliptic curve. (He also notes that the same result was established by Dieulefait, in the paper Existence of non-elliptic mod $\ell$ Galois representations for every $\ell > 5$.)