How do I install a headboard on an adjustable bed frame?

Suzanne: attach the headboard to the wall. Lots of ideas here:

https://www.bhg.com/rooms/bedroom/headboard/cheap-chic-headboard-projects/

Or... this was easy --- I merely hung an old window behind my bed (see photo)

some adjustable bed frames have a ' T ' on the headboard ends.

these can be used using bolts and drilling holes in the wood

headboard frame / legs.

Do you mean one of those beds that raise up and down? if so you will have to use a French Cleat to fasten the headboard to the wall.... here is a home-made one.... https://www.youtube.com/watch?v=DmLQkRDOgOE

You can also buy French cleats on Amazon rated to the weight of anything to be hung on the wall.... https://www.amazon.com/s?k=french+cleat+for+headboard&crid=2EZF1YUL67MRG&sprefix=french+cleat+for+hea%2Caps%2C260&ref=nb_sb_ss_i_1_20

Managing to attach the headboard can be a challenging task. Here are few steps to attach an headboard to the best adjustable bed.

1. Pull the bed away from the wall
2. Hold the frame where you want to adjust and lift it using a durable blocking. Hold the headboard temporarily by pushing a heavy object.
3. Slide a washer onto a bolt. Screw a nut until it is tight enough. Repeat same on other leg of the bed. Check whether it is hooked properly and then tighten the bolts using wrench.

Hope this helps you!