Having been a tennant and a landlord, I would say ask a maintenance guy about switching faucets. Be honest and tell them what you want. Offer to buy it if they will install it. If you do make a change w/o permission and a maintenance guy or apt. staff come in to service something and see a different faucet you could be in violation of some lease rule and asked to leave without your security deposit.
Plus, the top looks like a nice hard surface so drilling new holes may require special bits and more effort than it's worth.
Maybe a screw on sprayer like these would be agreeable to the GF.
Amazon.com: Faucet Sprayer Attachment by Miles Kimball: Flexible Faucet Sprayer: Kitchen & Dining
Danco 88270 360-Degree Swivel Aerator and Sprayer, Plastic, Chrome/White - Faucet Aerators And Adapters - Amazon.com
He brings up a good point, if one of my tenants wanted a different faucet and offered to pay for it, I would go and swap it out for them, keeping my faucet. But on the other hand if I went to one of my properties for a deposit refund inspection and their was some $70 home depot price pfister unit in place of the kingston brass units I put in every single one of my units
http://www.kingstonbrass.com/N_A_p/ks7718txbs.htm , I would be deducting the cost to replace it labor and matierials from their deposit. I use the same faucet in all {except for a few lav and bath faucets} my units, its easier to keep track of, I have parts to fix them with, and got a great deal on buying a few cases of them...
That kingston brass faucet in the link above cost me $88 each and I bought just under 40 of them {all they had in stock}, they have to weigh 10 lbs each all brass in a few finishes, that sprayer is super heavy duty, the connections are all brass fitted copper, the hose for the sprayer is nylon net over s/s rubber, I have been using them for a while and no issues as of yet, the finishes hold up well, they never leak, ect.. I like the mono deck mount because they install anywhere, if you have a 3 hole you use the deck plate, if their is just a single hole with no sprayer {this is getting popular}, I just drill a hole..
Even in granite the holes only take 5 minutes, with the new Milwaukee bits its even easier, it has a spring loaded center guide {you remove after you start the hole in about 3/8"}, all you need is a drill and water spray bottle, keep it wet {I dip the cutter in water every 30 seconds and clean out the hole}, rock the bit around a little once you get started and you will drill through a granite counter top in 4-5 minutes... I have done it a few times, I just did the pot filler hole in my own kitchen...
I also did a small cold only faucet for a new piece of granite I added to our kitchen with a long trough sink, I love this sink, fill it with icy water and beer, or fill it with ice and shrimp cocktails, clams, crab legs, ect for parties...
