I have a 2013 Trek Madone 4.5 with 10-speed Ultegra groupset, current crankset is a Shimano R565 50/36 compact. I want to replace the worn out crankset with a new lighter and up to date crankset. Will the new Shimano 105 FC-R7000 11 speed crankset work on this bike?
According to the latest Shimano Compatibility page the FC-565 and the FC-R7000 share compatibility with the same Hollowtech 2 road bottom brackets. So, as far as fit, you could drop in the FC-R7000. Be aware that the spacing of the 11 speed, 105, crankset will vary slightly from the R565's 10 speed spacing, and you may encounter some shifting hiccups from a wider 10 speed chain in use. The Ultegra front shifter and derailleur, designed for 10 speed systems but used on 11s design, may have a slightly different actuation ratio which may effect the smoothness of shifting in front. Shimano will not say these two crankset are replaceable with each other. Nor the ten speed Ultegra shifter and front derailleur with the 11 speed 105 crankset.
However, my experience as well as others' (with lots of documentation online) has been that with the front drive systems, moving up to the next generation chainwheel and/or derailleur will not cause significant problems. Things work just fine if set up properly. I currently run 10 speed chainsets on my mountain bikes, which have nine speed rear drivetrains. The hardtail has a 10 speed front derailleur (Deore FD-6xx) that is shifted by a 3x9 front shifter. Initially it moved the chain perfectly on a triple Deore (FC-M510) chainset marketed as 9 speed. I recently got a 10 speed Hollowtech 2 crankset (FC-M610) and using same shifter and a 9s chain, things are very smooth. My FSR Stumpjumper, still equipped with the 9s Deore components didn't miss a beat when I updated the crankset to a 10s triple (FC-M622--or something like that). Again, I continued using a 9s chain, new, as I also replaced the 9s cassette with the 9s XT model. Point is, up front, there's more flexibility to mix things up successfully.
With the rear drivetrain, especially in the higher speeds, 9 and up, there are a good many caveats to address before successful hybridization can occur and is seldom worth the trouble.
In your case, it's likely to work out just right, though I'd think about some of the suggestions in the comments. Especially note the 10s 5800 model 105 is a great value with very similar appearance to the 7000 105.