Yes and no.

RAW, you can gain proficiency in languages or tools through downtime training; there are more specific rules for this in Xanathar's. For skills, weapons, and armor, there are feats that grant proficiency and expertise (basically, double proficiency) in some things. Otherwise, your RAW options are just multiclassing. Certain classes get armor or weapons and rogue and bard both get a skill proficiency when you multiclass into them--plus rogue gets two expertise at level 2, and bard gets a class feature that gives them half their proficiency bonus on checks on which they are not proficient at level 2 and two expertise at level 3. There are also feats to allow even non-casters to get a couple cantrips or ritual casting.

(The main problem here, IMO, is that feats are now a replacement for increasing your stats, so martial characters are going to be choosing between STR/DEX/CON, actual combat feats, and versatility feats. I've seen some DMs get around this by giving out non-combat bonus feats, granting extra proficiencies based on starting INT, or by allowing training in cantrips and skills during long stretches of downtime.)

If you as the DM want to give out proficiency, you can. There are guidelines for this in the DMG. The DMG considers skill proficiency and feats to be a pretty high level reward, but you're the DM--you can do whatever you want.