Is it possible to update BIOS to UEFI?

It's theoretically possible, and may be semi-practical on some computers. What you'd need to do is to merge CoreBoot with a TianoCore UEFI payload. I've never tried it, but I've heard of this being done. Computer manufacturers could of course do this or something similar themselves, although I don't expect many will do so. Aside from the simple technical knowledge required to do this, the major obstacle is one of the size of the EEPROM chips on most motherboards. UEFI is pretty big, and although I don't know the details offhand, my understanding is that it's big enough that motherboard manufacturers wanting to support UEFI have had to increase the size of their EEPROMs to hold UEFI implementations. Thus, such an upgrade would work only on computers that had unusually large EEPROMs to begin with, or on computers on which you could swap out the EEPROM for a bigger one.

Another approach is to put TianoCore UEFI on your hard disk and boot it like a boot loader or OS. This is possible with a TianoCore build called DUET, which I wrote about here. This works only on some computers and is awkward to get running. I also don't know offhand if it supports Secure Boot. Even if it did, Secure Boot on such a configuration would be rather pointless, since it would still rely on a BIOS-mode boot loader to get started, so malware could theoretically still creep in though that unsecured foundation. (OTOH, I'm a bit skeptical that malware would survive the odd journey from BIOS through DUET to a booted OS.)

Gigabyte has also done it, for example http://www.gigabyte.co.nl/products/product-page.aspx?pid=3853#bios

It's definitely possible, and has been done.

MSI clickbios.

It's supposed to be a bit buggy, but functional nevertheless.