A more basic question is whether there even exists an algorithm to compute this number.
I've wondered this for a long time, and I honestly don't know what to expect.
Any algorithm would have to be quite subtle. In the early 1980's Shioda had to work quite hard to construct explicit examples of surfaces in $\mathbb{P}^3$, defined over $\mathbb{Q}$, with Picard number 1.

Added: Of course, I should have said Shioda's example have degree >1. As further
evidence of subtlety of the problem: one can decide whether an elliptic curve $E$
has CM by computing the Picard number of $E\times E$.