I created a bard and started a game, but I have a problem. My bard can't scribe any spells in her spellbook. She has high intelligence, and even have a few slot in her spellbook already, but when I open a scroll, there is no option to write it, while my mage get the option, is that a bug?

Why is it that everybody has BUGS today?

A Bard not being able to scribe spells is bizarre and, if true, is most definitely a bug.
Are you sure you're not given the option to write the spell because it happens to be one you already know?
For good luck, take your armor off before trying to scribe. It shouldn't make any difference, but it couldn't hurt.
Donate lots of gold to every Priest of Mystra you come across.

I have no spells learned at all. My character was never able to scribe spells, but my mage in the same game can. I tried with and without the armor.

But don't worry, I used Sword Coast Keeper to give my Bard all the spells she would need. ;)

Isn't the spell learning table different for bards? Don't they learn their first spells later than mages?

They do, but my bard had a few slots open aleady.

They do get spell slots later, but they should be able to learn a spell.