No. The Russian word бистро (bistro) is an adverb meaning "quickly," "fast," or "rapidly." It does not mean "hurry" (noun/verb).
The French word bistro (small restaurant) likely comes from Russian bystro — allegedly shouted by Russian officers in 1814 Paris demanding fast service. The French noun bistro was then borrowed back into Russian as бистро (café), but the adverb быстро remains "quickly."
So: быстро = quickly; спешите = hurry (command).