The smallest unit in a modern computer is the bit: it can either be 0 or 1, off or on, however you want to call its two states. That is: every modern computer works in base 2 (for integers; floating point numbers complicate things bu still use bits).
But displaying groups of bits to humans (even programmers) is tiresome, so some more convenient representations are chosen when showing those bit groups: the most common are base 8 (octal), base 10 (decimal) and base 16 (hexadecimal). Still, the computer does not know about hexadecimal, it's working with (groups of) bits. We call a processor's preferred group size of bits a word. For example, a 16-bit CPU prefers to do most or all of its operations on groups of 16 bits; no more, no less.
So the computer is working with base 2 numbers only. When you want to show the bit group to humans as a readable number, you need to format it. Which base you're using for the representation doesn't matter, this transformation from the internal value to a human-readable text representation is always done by the processor (CPU). Even when you want to show just the bit groups (base 2), some transformation to a text needs to be done.
To be more precise, this transformation from internal representation to text is done by software running on the CPU.
The mainboard/motherboard's function is to connect the CPU to all the input/output peripherals. Even though nowadays some parts of it do very complex operations/calculations, in the end it's not really taking part in the formatting of numbers other than providing a connection from the CPU to the graphics processor, which is going to display it, or some storage to save the representation.
So the complete answer to your question: "Which part of a computer does the conversion between binary and hexadecimal?" is: software running on the CPU, creating a representation that is passed to the graphics processor (usually via the mainboard) so you can see it on screen.