Btukfyl

My sense, from my own experience, is that people usually use the informal "in the morning" when using the casual "quarter to three" in words, and the more formal, technical-sounding "a.m." when using the precise "2:45" in numbers. If you're only mentioning an hour, of course, then there's no difference when speaking aloud, and either might be used.

However, while I say "usually", I suspect that that is not by a significant margin. There's nothing weird or unnatural about "quarter to three a.m." or "2:45 in the morning". Using both about the same time in a single phrase will make you look/sound silly, though. Don't say "2:45 a.m. in the morning"¹. "From 9:15 in the morning to quarter to five p.m." is perhaps idiosyncratic, and might draw odd looks in a formal context, but is in no way wrong.

¹: There will likely be edge cases where it's appropriate to say that, but it will arise naturally from the surrounding text - it's likely that, in most analyses, they wouldn't actually be the same phrase. "I'm leaving at 7 a.m. in the morning" could be better rephrased to avoid the apparent redundancy, in the case where it means "I'm leaving at 7 a.m. tomorrow". However, if you parse the apparently-redundant sentence with both "at 7 a.m." and "in the morning" as separate adverbials of time, you can see that in the morning and a.m. aren't part of the same phrase.

From my initial comment, I personally find the juxtaposition of "colloquial" quarter to [hour] and "formal, official" p.m. quite jarring in the example as given. I'd much prefer either It's quarter to five in the afternoon or It's four forty-five p.m. (or go the whole hog with military time; It's sixteen forty-five).

But that's very much a personal opinion about style. As regards using both a.m. / p.m. (capitalised on not, with periods or not, according to stylistic preference) and in the morning / afternoon / evening, consider...

"at 7 am in the morning" - about 4510 written instances in Google Books

...where obviously there will be plenty more for the actual word seven, or for other times. I believe it's relevant that in many (but by no means all) those hits, in the morning will mean tomorrow morning. In which case it probably wouldn't even occur to most native speakers that there was any tautologous repetition at all (it simply clarifies the day of the specified time).

There's often also an element of (perfectly idiomatic / natural) emphasis in my example (either or both of a.m. and in the morning could be seen as implying [unusually] early. This isn't so likely with p.m. + in the afternoon, so you won't come across that collocation very often.

But the justification for "clarification / emphasis" certainly re-appears with...

"at 7 pm in the evening" - about 3960 hits in Google Books

Perhaps I should have switched to six p.m. there, to add weight to my point here. Suppose you hear one work colleague say to another, I'll see you at 6 p.m. in the evening. In context, it's extremely likely the speaker expects himself and/or the other person to return home after work before they meet again. Effectively, in the evening is an "additional clarification" element, used to distinguish "working afternoon" from "leisure time evening".

So as you should be able to see, there are contexts where it's absolutely fine to include both "time of day" elements. Arguably it's "clumsy" to do this unless you actually want either the nuance of "emphasis" OR the other associations (morning = early, evening = after work) involved.

Since it might be difficult for non-native speakers to reliably identify which contexts do justify the repetition, and it's undoubtedly true that there will be other more "neutral" contexts where it's noticeably "awkward" to include both, my general rule of thumb would be to avoid it. But if you think you have a context that seems to match the "valid" constructions I've referred to above, don't rule it out. And certainly don't feel you can justifiably criticise any native speaker who does it!

You can use either A.M. or in the morning, and P.M. or in the evening. It is incorrect to use both A.M. and in the morning in the same sentence.

In your case, you would be correct in saying It's a quarter to five P.M.

Yes. No problem in that usage. Make sure both a.m. and p.m. are properly written in lowercase with periods.

I have found an example sentence from NY Times website:

Over the past two weeks, however, a group of hooligans have made a routine of partying on my roof at 3:30 a.m.

And here's another one from Slate, showing something more akin to your example:

No prisoner is allowed under any circumstances to leave his cell after half-past five p.m.