Inspect the mattress first. Look at its life expectancy date. In general, mattresses are usable for a seven-year period, after which time they begin to "thin" and sag. This causes less padding to be available, hence less sound deadening/dampening. While this may not be the source, it will exacerbate the squeaking noise.
Check the link spring. Remove the mattress and sit and/or lie directly on the link spring. Listen for the squeak. Once detected, press on that area several times to ensure you have found the source of the squeak.
Spray the link spring with a lubricant to remove the squeak. Then press down to test. If the squeak remains, look at where the link spring attaches to the frame. If it has shifted, it might result in a squeaking sound from improper alignment.
Move the frame itself. With the mattress and link spring off the day bed, attempt to move the frame back and forth to find the squeak. If you hear the squeak coming from a frame, simply spray it with lubricant to stop the squeaking from occurring.
Inspect the mounting brackets and screws attaching the head board and the foot board. If one is loose, tighten it with a wrench, then spray it with lubricant to stop the squeaking.