I conjecture that there are infinitely many correct solutions to this equation: (Where we are assuming $a,b \in \Bbb{N}$) $$a!+1=b^2$$ I chose to list the first three solutions below:
$4!+1=5^2$
$5!+1=11^2$
$7!+1=71^2$
and so on...
Could I have a proof or disproof of my conjecture? (I don't even know where to start)
This is known as Brocard's problem.
It has been shown there are only finitely many solutions given the ABC conjecture is true (this I believe was proved recently), and it is conjectured the 3 solutions you have given are the only solutions. Calculations up to $n=10^9$ show no other solutions.