Presumably, the child was conceived naturally in this scene in Chapter 18 of Maximum Ride Forever:
Fang sighed. “Maximum Ride, you’re the most stubborn person I’ve ever met, and sometimes it seems like your sole purpose in life is to make mine harder, but I swear, I love you more than I thought I could love anyone or anything.”
“Then stay,” I whispered, clenching my eyes shut just as they started to well up.
I felt Fang’s hands on the sides of my face, his thumbs wiping away my tears. I felt the heat coming off his body, heard the catch of his breath. And when our lips finally came together, our kisses were urgent, our bodies hungry. As he moved his hands through my tangled hair, I looked up at him. Inhaled. And said, “Yes.”
We sank to the ground, the dried leaves crinkling under us, and time fell away for a while.
I couldn’t tell if our voices rose in pleasure or pain, couldn’t tell if my heart was breaking or bursting open with joy. I only knew I didn’t want to pull away from him for a single second, and it was only when we both gasped for a breath that I realized Fang’s eyes were squeezed shut and his lashes were wet.
Then, towards the end of the novel it's mentioned that the child, Phoenix, is born. Although the birth is discussed in a past-tense manner
It wasn’t all bad, though. In the room I shared with Fang, I swore through a much-too-long childbirth and may have punched Fang and ripped a pillow in half, but I ended up with a wrinkled, utterly perfect nugget of joy as a souvenir.
We see that it was a live, natural birth, and did not involve an egg, despite my presumption and Max's fears:
I patted my belly button, feeling the swell that was growing a tiny bit bigger every day. I pressed my knuckles against the small curve, kneading in, but it always rebounded.
I really hoped this wasn’t going to be a great big egg to lay. How could I possibly sit still on it for nine months?
As far as their ages go, the last bit of information I can find is than in the book Fang, Max turns 15. Fang is 6 months younger than she is. I can't find information about how long after each novel then next one takes place, but I believe they're generally immediately following. Max was 14 in the first novel, and it took until the 6th for her to have a birthday.
While it may be one's assumption that the two characters were underage, you must remember that they are not human, as they're human-avian hybrids, among other things, and that by this time the apocalypse had struck, during the conclusion of Nevermore, so there wasn't even a legal system to classify "underage".