No.....Elder
PS: According to what I've experienced and read the ripening process ends at the moment the fig is picked from the tree.

Figs do not continue to ripen off the tree, nor do they respond to ethylene treatement for ripening. Having said that, however, I am confused about the labels "tree ripened" and "mature" on the following UCDavis website.

~james

Here is a link that might be useful: Cultivar Pictures

My understanding is that figs do respond to ethylene treatment while they are on the tree. This is the basis for oleification; olive (and other oils) produce ethylene upon decomposition. I can refer you to scientific papers on the subject if you are interested.

I've not read that they "ripen", but I have read that you can keep them a few days and that that they will "soften".

It is not the first time that I harvested figs and left
them a day or two inside (some continued to ooze honey
from the eye) and I think overall, got better a eye
appeal or maybe better tasting?

Any figs that I have grown, I have noticed that "matured" is a oint at which they have gotten to fheir fullest poenial in size and have somewhat of a juicyness inside of them. If you leave them on the tree once they have "matured" they will ripen further getting more juice inside of them until the "nectar" starts to ooze from the "eye" that is when they are "tree ripened"

Last year I had around 3 pounds of figs on the tree that were not completely ripe, They changed color, still a little hard. It was late september and we were expecting freezing temps at night. So I picked them and put them in a ziplock bag, I left it on kitchen counter for 2 days. Surprisingly, they got soft and were sweet. Talking about Ethylene gas, there were a couple figs that were already ripe in the bag, they must've released that gas and ripenned the unripe figs.
Give it a try.

Bass

In my experience, they WILL soften and get sweeter when you let them sit at room temperature for a couple of days. The books aren't always right, do you own experiments