The answer is yes. There's a version of Weil conjectures for all algebraic varieties, though you have to be a bit careful (if you're not projective
I don't know what Weil's orginal proof was, but proving the Riemann hypothesis is a bit different).
However, Weil conjectures for P^2 is extremely easy, it's pretty boring. If just connects the fact that since the cohomology of P^2 is 1 dimensional in degrees 0,2,4 to the fact generated by algebraic cycles. You just note that there are 1+q+q^2 points in P^2 .
For a good introduction, you can read J.S. Milne's notes on etale over F_q (which corresponds to the cohomology being one dimensional in degrees 0,2,4).