Correct spelling, explanation: months is a plural noun and it’s used very often in the English language. It consists of two parts: the base, which is month, and the suffix, which is -s. There is no reason why the ending should change to -es. It is worth remembering this when spelling this noun. Monthes is an incorrect form.

Definition of months:
noun (plural), twelve named periods into which a year is divided
I saw my ex-girlfriend two months ago.
It took him several months to finish his BA dissertation.

Collocations with months:
Some most commonly used collocations include: [one, two, three…] months later, and several months.
The news of their marriage was not made public until three months later.
Her most brilliant work was done during several months of relative inactivity.

Incorrect spelling, explanation: this error can be easily explained. When we create regular plural nouns in English, we add one of the two endings to the singular forms. It is -es or -s. If a word ends in –s, –sh, –ch, –x, or –z, you add –es. For almost all other nouns, you add –s to pluralize. It’s not difficult to mistake one suffix for the other. In this case, months is the only correct version, not monthes.