2005 Infinity Q45 driver's side headlight is out

Pull the headlight housing out and inspect the wire harness, the connectors, the bulb sockets, and electrical grounds. If all that is good start tracing the problem back along the wire harness.

Chances are its just the bulb. I can't really see, but if you have the multi-lens projectors (a circle with 5-6 little lenses in it, kind of like an insects compound eye) then it's a D2S HID bulb. You can test by swapping the bulb from one side to another, if the headlight that is currently not working stays not working with the bulb from the other side installed it will be the HID ballast, which is a dealer only part. If it's just the bulb that is out I suggest buying a new quality set, and not ebay junk as they don't last very long and have horrible coloring. 'The Retrofit Source' makes a very affordable, but good quality replacement D2S bulb (google retrofit source), I would either use their bulbs, or go with the oem bulb if budget is not a concern. Replace both bulbs at once, believe me the other one isn't too far from burning out either....

I have a 2002 Infiniti Q45. Both your auxiliary parking light bulb (194 or 168, located towards the top of the housing between the low beam bulb and halogen H1 high beam bulb) and your high intensity discharge D2S low beam bulb are out. Any replacement of the bulbs will require removal of the entire headlight assembly. If the ballast needs to be replaced, it is mounted at the bottom of the headlight assembly. Photo is for reference only, displaying a model with aftermarket (5000 kelvin color temperature) parking light bulbs and aftermarket (3000 kelvin color temperature) fog lights on.