You're both wrong. When referring to a word, letter, or number by name, it should always be enclosed in quotation marks. So the correct form is:

There are two "b"s in "Bobert".

That, of course, assumes that the entire phrase itself isn't a direct quotaion. If that were the case, the proper form would be:

"There are two 'b's in 'Bobert'."

In such a construction, the mark after the "b" is a single quotation mark, not an apostrophe. They just look the same.

To further complicate matters, I'm old-fashioned and always use full quotation marks, but some writers have opted for the single quotation mark style. In that case, the proper form would be:

'There are two "b"s in "Bobert".'

Since the writer is enclosing the entire quotation within single quotation marks, full quatation marks are used for the "b".

Now go back to bed until your back feels better.