You could start counting from ANY position. It will be a draw if 50 moves elapse without a piece being taken or a pawn being moved. You could even start counting from the opening position - both sides could just move their knights around randomly, and maybe move their rooks to where the knights used to be, for 50 moves.
The fact that the pawns are blocked in the position you show doesn't matter to the 50 move rule, except that it makes them less likely to move and thus reset the count. Whether a pawn could move or a piece could be captured is irrelevant - what matters is whether that actually does happen in the game.
Although you can start counting, White has several ways to reset the count in the position you show. They could capture the remaining pawn, and thus reset the count to 0. With the pawn out of the way, they could later move their own pawn several times to keep resetting the count. They could even move their own queen next to the Black king to force it to be captured, to reset the count - remember, ANY capture resets it.
There is a USCF rule (14F4) that someone in sudden death with less than 5 minutes left, and a position where captures and pawn moves are unlikely, can ask the TD for assistance in counting moves. Whether the TD agrees is up to his discretion. In such a case, the pawns being locked might factor in to whether he agrees to help.
(I'm ignoring that there's no White king in your diagram; I assume that to be an oversight and not part of the question. Lack of a king means there's an ongoing illegal position, and that has its own rules.)