Actually, "Maybe I('ve) got to wash the apples before I eat them". Which is the same as "Maybe I (have) got to wash the apples before I eat them". We usually just say "I've got to" or, more informally, I've gotta

gotta = got + to

"I've got to wash" = "I have to wash" but the second one is more informal.

All these differ somehow:

-I should wash. (I have to do it, but it is not as important) "I should go to sleep, but I want to see the last episode.

-I must wash. (Biggest importance) "I must study for the test"

-I have to wash/I've got to wash. (It express lower importance than "must" but more than "should")

-I'd better wash. (I'd = I + had. It implies negative consequences) "I'd better wash the apples, or I can get sick" 

I've been having troubles with this phrase (It's something a friend wrote)
"Maybe I got to wash the apples before I eat them"

Or is better to say "Maybe I have to wash the apples before I eat them"?

After "maybe," I would generally use "should" or "ought to," since "have to" expresses obligation or necessity -- no "maybes" allowed.

"I have to wash" can also be said "I've got to wash," they're equivalent. It's usual in conversation (at least in North America) to omit the auxiliary "have" and say "I got to wash," but this is considered poor diction. In this context, North Americans pronounce the "t" similar to a short Spanish "r" -- "I gotta" /aigárə/

Note that "I got to wash" could also be interpreted as the simple past of "I get to wash," meaning that I had the opportunity to wash, but then the second half of the sentence would also be in the past "I got to wash the apples before I ate them." You could also use "I managed to wash..." or "I was able to wash" with the same meaning. With this meaning, the "t" sound is pronounced as a "t" even in American English.

Your second sentence is correct: "Maybe I have to wash the apples before I eat them."

Other options:

"Maybe I need to wash the apples..."

"Maybe I should wash the apples..."