The state capital of New Mexico is Santa Fe. Santa Fe is not the most populated city in NM, that would be Albuquerque, but it is the oldest.
The rather unremarkable building you are looking at on the left is the Palace of the Governors. I know, it doesn't look like a palace. It was built by the Spanish in 1610 and was the seat of government in NM for almost 3 centuries. It is the only capital in the US to have housed the government of 4 different nations, Spain, Mexico, the Confederacy and the US. It is also the oldest continually occupied building in the US.
It is still there today on the Santa Fe plaza and as you may be able to tell from the photo Native Americans set up in front every day and sell native jewelry and other items. Inside it now houses the New Mexico History Museum.