Since it's not actually a question, I'm not sure if a question mark should be used or not.
For this type of conversation, I can't tell you. It's so informal by nature that the parties only need to be able to understand one another.
The person is probably writing it with a question mark because the intonation in their head was rising, indicating a question.
At the same time, it may well actually be a question, which I'll explain in a bit.
I know if you were to greet and ask a question of someone before their
response, you'd use a period like this: "Hey, Matt. What are you
doing?," but I'm not sure what punctuation to use in this instance.
This is the beauty of English; there are several ways to punctuate that, some leaving the sentence in question as statement, while others will show that it is a question. For example:
Hey, Matt. What are you doing?,
This is fine. One greeting and one separate question
Hey Matt, what are you doing?,
Fine too. Hey Matt is a greeting or an interjection/greeting, so you can make it all one sentence (Hint: Hello is an interjection).
If you're just writing Hey [Hello] Matt, then it's not really a question and would require an exclamation point or a period.
This is such informal writing that anything you choose should be fine.
One final word, as I said above, many people do this because they want their written words to sound as if they were speaking them. So sometimes question marks are used, as I've said, to signal a rising pitch toward the end. One might do this if the statement is ambiguous or can be misconstrued.