Automatically Through Software Update

It is likely that OS X will automatically prompt you to download and install any recommended firmware update. This happens as part of OS X's software update process.

See About firmware updates for Intel-based Macs for the specific update steps on older Macs:

Learn how to locate, download, and apply a firmware update to an Intel-based Mac computer.

All about the Mac Pro,1.1.

Manually via the Apple Support Site

Alternatively, you can find firmware updates on the Apple Support site. For example, this update appeared soon after the Mac Pro1.1 was released in 2006: Mac Pro SMC Firmware Update 1.1:

The SMC Firmware Update adjusts fan behavior in the Mac Pro.

Download and run the updater. The updater will check and prompt you if it is required.

These firmware updates are effectively the same as a PC BIOS. As a Mac user, there is nothing to manage or update at the BIOS level; it is controlled by Apple.

This post at the macrumors forum lists which CPUs are capatible, outlines the various issues with upgrading, and has links to detailed guides on the upgrade process, including the firmware upgrade.

http://forums.macrumors.com/threads/mac-pro-cpu-compatibility-list.1954766/

Read everything through before starting and you should be fine. I spent a lot of time making sure I had it all clear in my head before buying the processors and starting the job. Double check everything and ask questions(I recommend the same site again) if you are unsure about something. Drop me a line if you are having problems finding a particular guide and I will see what I can dig up and upgrade the answer here if relevant. I used lidded CPUs and did the 'washer stack' method mentioned on the site to upgrade my 4,1 with few problems following the guide on how to tighten down the heat sinks. De-lidded CPUs cost quite a bit more and are not worth it IMHO, or worth the risk of de-lidding them yourself. YMMV.

The firmware upgrade to a 5,1 is very simple and required to use Westmere processors. It is the easiest part of the upgrade. In fact you can do it all by itself without upgrading anything else with the system and you computer will for a practicable purposes be identical to a 5,1.

I upgraded my system last year with two Westmere four core Dual Xeon X5677 3.46Ghz CPUs from two Nehalem four core E5520 Xeon at 2.26Ghz. I decided that the additional cost of going to hex core processors was not worth it to me for the work I do. It was over twice the cost for used hex core processors at the time and they would have had a slightly slower clock speed. Your needs may differ and I'm sure that prices now are very different. Seeing that you are on a single processor system, a hex core may make more sense for you as you wont have to double the CPU cost by buying two as I had too. Watch the prices for a few weeks to see where the market is at and to get the best pice performance ratio that fits your needs.

Also consider upgrading your Video Card if you haven't already. There is a post at the same site with a guide on what PC cards work and which don't with work arounds and any problems or limitations. It is very long, but worth digging through.

http://forums.macrumors.com/threads/frequently-asked-questions-about-nvidia-pc-non-efi-graphics-cards.1440150/

Another tip for upgrading is to look for used, registered server RAM. Registered RAM must must all be the same type; you can't mix it with unregistered DIMMS. Normally it is more expensive, but because the type of DIMM the 4,1 and 5,1 uses are quite old now, not being able to mix it, and the motherboard must specifically support registered RAM(which the Mac Pros do, most consumer machines don't), you can find it really cheap from people parting out PC servers on ebay. I can't remember the price, but I upgraded to 24GB(six 4GB 1333 DIMMs) for not much money at all.

The forums mentioned have loads of information and the people are willing to help, just be sure to read up about the process as much as you can before posting a question. Like many forums out there, they will jump on you a bit if you didn't take the time to read the available information beforehand and jumped straight to asking simple questions. Make yourself informed. Don't be put off, this is a doable upgrade and has brought a lot of life to my old system. This era of Mac pros were so well designed and I love to see their life extended. If you really get lost after spending some time gathering information, drop me a line and I'll see if I can point you in the right direction. Good luck.