It's possible, but very, very unlikely. As in, not in our lifetimes. :)
Oxygen is very electronegative (electronegativity of #3.5#), and after water gets protonated and becomes hydronium, it becomes less stable than water. Try to add another proton, and it'll become too unstable. I'm not exaggerating when I say that probably a million times to one, or more, the proton would get rejected.
Now, if you protonated every single water in solution, there'd be a better chance, but that's almost impossible to accomplish---the concentration of pure water is actually about #"55.34 M"# (it is its own "solvent" in that case). That's so high that we often don't even mention that that's the true concentration of water by itself.
I personally haven't heard of an acid that is strong enough to dissociate and achieve a concentration higher than #"55.34 M"#... if there was such a thing, that would be rather miraculous (and impossible to store in a hood without fainting/dying from the fumes first---#"12 M HCl"# is strong enough!).
What could happen though is that one proton coordinates with a cluster of water molecules in solution, so you could get:
#H^(+) + 2H_2O rightleftharpoons H_5O_2^(+)#
#H^(+) + 3H_2O rightleftharpoons H_7O_3^(+)#
and so on, but we often just say, for simplicity, #H^(+)#, and other times, we also say
#H^(+) + H_2O rightleftharpoons H_3O^(+)#
But no, there is practically zero chance that #H_4O^(2+)# can occur.
