"Quite an X" is an idiom meaning a very extreme or unusual example of X, usually in a positive way, but depending on context it could be bad. Like if you say, "Wow, that's quite a car!" you mean that it is a very good car. That might mean very fast, very luxurious, etc, depending on the context.
If someone says, "That's quite a problem you have there," that would be a bad thing. It is a very serious or perhaps unusual problem.
"We're quite an institution" means that the institution they're referring to is a very good and great institution. From the context you give above I think they are stressing the longevity of the capitol building, though a larger context might indicate that they're talking about some group that is organizing the celebration.