Both sentences are correct and sound natural, but they convey slightly different tones: 1. "Can I boast a little bit?"
This sounds more polite and self-aware. It acknowledges that you want to talk about your achievements but in a way that doesn’t seem arrogant. 2. "Can I show off a little bit?"
This is more casual and playful. It can be interpreted as lighthearted and fun, depending on the context. Which to use? * If you want to sound modest: "Can I boast a little bit?" * If you’re being playful with your friend: "Can I show off a little bit?"

Hi Anzhela! Either are fine, or you could use the popular idiom: to blow your own trumpet, which means 'to show off what you can do'. For example: "Well, to blow my own trumpet...." (and list your achievements) Or, what would be even better, you could use it in the English-style 'passive' sense (where we don't say exactly what we mean, in order to be more diplomatic): "I don't want to blow my own trumpet, but..." or "I hope I'm not blowing my own trumpet here, but..." (and then list your achievements). And, not to blow my own trumpet, but actually I recently recorded a Podcast on this exact topic, if you have a look at my profile you'll find it ;) Why the British don't say what they mean: https://www.italki.com/podcast/episode/4l19gbgyrjxpxj5pug2szj

To say "Can I boast a little bit?" or "Can I show off a little bit?" You can say this to make it be more colloquial and soft. "Can I toot my own horn?"