Pulsar's answer is indeed correct, but let me expand a bit more.
What happens when a gas giant shrinks?
A uniform mass will have a self gravitational potential of $-\frac{3GM^2}{5R}$. If we decrease its radius, its potential will decrease as well and the difference will be turned into thermal energy. Although gas giants and stars are not uniform mass balls, their gravitational binding energy is still proportional to $\frac{GM^2}{R}$, Thus if the radius decreases it will release energy, which will raise the temperature in return.
What happens when the temperature increases?
Assuming the gas in those planets obey the ideal gas law $$PV=nRT$$ (where $R$ is not the radius but the molar gas constant $R=8.314\,\text{J K}^{−1}\text{mol}^{-1}$), it's obvious that when $T$ increases and $V$ decreases (due to the shrink in the previous section) $P$ must increase. Note that most real gases behave qualitatively like an ideal gas, so this is not a crazy assumption.
So what is the big picture?
The planet shrinks a little bit, the potential difference turns into thermal energy and its temperature rises. The rise in temperature will cause the pressure to rise and prevent the planet from shrinking further (holding the planet in hydrostatic equilibrium). However, the planet also loses energy due to EM radiation as well, so it will continuously shrink and radiate. The process is called Kelvin–Helmholtz mechanism.
For instance, Jupiter is shrinking the tiny bit of $2\,\text{cm}$ each year. Although you might think this is really nothing, the amount of heat produced is similar to the total solar radiation it receives.
Addendum (Nov. 2020)
As Rob Jeffries has correctly pointed out, what ultimately keeps a gas giant from collapsing indefinitely is the electron degeneracy pressure. Eventually because of high pressure the hydrogen and other elements in the deep interior of the gas giant will undergo a phase transition to a metallic phase and will not compress any further.
Red giants and asymptotic giants have some close similarities, and one actually evolves into the other. Both have an extended envelope of relatively cool, non-burning material (mostly $\rm{H}$, $\rm{He}$). They also each have a core of dense, non-burning material; in the case of the red giant this is mostly $\rm{He}$, while for the asymptotic giant it's $\rm{C}$ and $\rm{O}$.
The burning shell in the red giant is $\rm{H}$. For stars of the right mass, the conditions (density, temperature) in the core will periodically be sufficient to ignite the $\rm{He}$ causing a "core flash".
Red giant structure:
After the red giant branch of stellar evolution there is a brief period where the $\rm{He}$ core burns called the horizontal branch. Once the He core is exhausted (it's been converted to $\rm{C}$ and $\rm{O}$), the star starts on the asymptotic giant branch. This branch has two parts, the early asymptotic giant branch (E-AGB) and the thermal-pulse asymptotic giant branch (TP-AGB).
E-AGB structure:
Stars on the E-AGB are like red giants, but in addition to a $\rm{H}$ burning shell there is a $\rm{He}$ burning shell (the energy output is dominated by the He burning shell). In the TP-AGB, the $\rm{H}$ shell picks up again and dominates the energy output, but periodically as the $\rm{He}$ produced by $\rm{H}$ burning is accreted onto the $\rm{He}$ shell, "helium shell flashes" occur, analogously to helium core flashes in red giants.
Source/Reference: Carroll & Ostlie "An Introduction to Modern Astrophysics: 2nd Edition" (Pearson)
Best Answer
If you could make it down to the core, then yes, you would probably be able to experience a "solid surface" (where I put that in quotes for reasons that should be apparent in a moment).
The question really gets to, though, what you consider to be a "solid" and a "surface" in a gas giant. The issue at hand is what the Wikipedia article stated - as you go down through the atmosphere, you encounter denser and denser material. Gas under incredibly high pressure will start to behave like a liquid, and be as dense or denser than a liquid such as water. There would be no definite point at which you could say the stuff above you is clearly "air-like" while the stuff below you is clearly "water-like," it's a gradient.
You would also get crushed long before you made it anywhere near the core, just like it's only fairly recently that we've been able to build submersible vessels that can go to the deepest parts of the ocean on Earth.