We tried grilling apricots once, thinking that they would be like grilled peaches. They were most definitely not like peaches! Grilling them made them super sour and acidic rather than super sweet and delicious. Definitely do not recommend grilled apricots, but grilled peaches are still yummy!
We don't use a recipe for them, we just throw them on medium indirect heat until they get a bit of char. It makes them much juicier, and it makes the peaches sweeter!
Ah, I think that apricots don't have enough 'free' juice in them to grill them dry like that. If you use slightly unripened apricots (halved and pitted) you can stick them in a sweet marinade for no more than an hour that isn't too heavily flavored and grill them with the cut side down. It goes really well with soft and gamy cheese, and makes a nice side for grilled lamb or goat.