Learn Portuguese (Portugal): Eu peço a conta ao senhor.

Peço is the "I" form of the verb pedir (to ask). This verb is irregular in the present tense. To create a soft "s" sound in this specific form, the spelling changes to peço. The other present tense forms are more regular and use the letter d, like pedes (you ask) and pede (he/she asks).

Why use peço here instead of pergunto? Don't both mean "ask"?

Yes, but they are used for completely different things. You use pedir (and its form peço) when you are asking for an object, an action, or a favor. You use perguntar when you are asking a question to get information. Since you are asking for a physical thing (the bill), peço is the correct choice.

In English we say "ask the gentleman", so why does the Portuguese sentence use ao senhor instead of just o senhor?

In Portuguese, the verb pedir is structured a bit like the English word "request" or "hand". You request a thing (a conta) to a person. The preposition a (to) combines with the word o (the) to make the contraction ao. Literally, the sentence translates to "I request the bill to the gentleman."

Could ao senhor also mean "to you"?