It is always important to remember that English, like many other languages has complex expressions that often challenge grammatical explanation.
Now, there are two approaches to the analysis of a sentence:
1. The identification of parts of speech in the sentence. So, if the above question is, indeed, literal: "Identify the verb in this sentence," then the only verb is "come." Out in this instance is defined in English grammar as an adverb (though it is versatile and has other grammatical uses too), and falls into a "lexical word class." This definition can be verified in complete dictionaries and English grammar textbooks such as those published by Prentice-Hall and MacMillan, used throughout the United States in public and private schools. As a note, here is the definition of particles:
particle: a word or a part of a word that has a grammatical purpose but often has little or no meaning: In the sentence 'I tidied up the room', the adverb 'up' is a particle. (Cambridge Dictionary)
technical: an adverb or preposition that can combine with a verb to form a phrasal verb (Longman Dictionary)
Grammar: a minor function word that has comparatively little meaning and does not inflect....
(in English) any of the class of words such as in, up, off, over, used with verbs to make phrasal verbs. (Oxford English Dictionary)
2. The identification of syntax in the sentence; that is, how are the words used--what are the subject, predicate, direct, indirect objects, prepositional phrases, phrasal verbs etc. The predicate of the given sentence, They come out only in the summer months, is, then, "come out."
This phrase, "come out" is considered a verb phrase because it is a verb plus an adverb. For, as already mentioned, the word out, by itself, is an adverb that can function as a particle. When used with the verb come, it then becomes part of a verb phrase. Verb phrases (phrasal verbs) may be formed with a verb plus an adverb or plus a preposition or plus both. Dictionaries list many phrasal verbs beginning with come (they also list a few figurative idiomatic expressions with come).
phrasal verb (grammar): a combination of a verb and an adverb or a verb and a preposition, or both.... (Cambridge Dictionary)
We have to take the analysis of the predicate verb one step further, though. "Come out" is an action phrasal verb. It is not a linking one. "Come out" isn't reflecting what "they" are. It reflects what "they" do. Substituting "flowers" or "insects" for "they" shows that "come out" reflects how the flowers grow or how the insects live. It doesn't link to what the flowers or insects are (as in "pretty" or "small"). As a result, "in" is analyzed as part of the phrasal verb "come out in" with "the summer months" analyzed as the noun phrase filling the object function. Objects can be filled by nouns, noun phrases, adverbs and adjectives but not by prepositions, like "in," nor by prepositional phrases, like "in the summer months."
To finish the analysis, "only" is analysed as an interjected adverb that can function in one or two other places just as well, for example: "They come out in the summer months only."
A better question might be “identify the predicate in this sentence.” A sentence contains a subject and a predicate; verbs and nouns are parts of speech, not parts of a sentence. Having said that, however, this sentence contains what is known as a “verb plus particle”; that is, the predicate of the sentence is “come out”. This kind of construction is more common than you would expect, partly because a lot of these verbs plus particles in Latin are combinations of verbs plus prefixes (e-merge – merge from, over-whelm, etc.). In this sentence, then, the “verb” is “come” – an action word, with the adverbial particle (how? “out”). Many particles are prepositions, and the predicate is then called a “verb plus prep” or “verb plus preposition” -- “give” (in), “carry” (on), etc. George Bernard Shaw’s famous objection to the “rule” of not ending a sentence in a preposition (as in “Where is the address at?”) shows the value of verb-plus-prep combinations: “That is a proposition up with which I will not put.”
adverbial particle: an adverb used especially after a verb to show position, direction of movement, etc.
In ‘come back’, ‘break down’ and ‘fall off’, ‘back’, ‘down’ and ‘off’ are all adverbial particles. (Oxford Dictionary)