See other months on Apple Watch Calendar?

I don't believe you can. However, as others have suggested in the comments, there are third party apps you can use that will do the job. I suggest using Sunrise Calendar. You can integrate it with your iCloud and Google calendars (and I believe some other calendars), and it has a day-by-day view on the Apple Watch that works nicely (not your typical calendar view, but it works, showing every day of that month and other months). If you want to see the days in the next month, just keep on scrolling down and tapping "Load more" until you are at the desired month. You can't see each entire month, but this way, you can at the very least see every single day of many months in the future, including the days on which you have no events, which is the feature I'm assuming you're looking for.

UPDATE: Over time, I've found a few other apps that may help, though there are none that show you the entire year, month by month, at a glance.

Week Agenda: You can easily view each day of multiple months with this app, but you'll have to tap on arrow keys to scroll through all of the days.

Add Event: Though to create reminders/alerts and repeating events, you'd need to buy an in app purchase, this app will let you add calendar events to the default calendar on your phone for free. This is not what you were looking for, but it may prove helpful given that your question was in regards to calendars.

I have seen plenty of apps that replicate the functionality of the original Calendar app that comes natively with the watch, but they only show one month at a time.

If you're willing to deviate from apps that are meant for calendaring, another solution would be to use something like Exhibition, which allows you to view selected photos on the watch. If you have a square photo of the desired month(s) on a calendar, you could always put them on the app and view them on your watch that way.

UPDATE 2: Now that Sunrise has been discontinued, another alternative you can look into is Coyomi. The little calendar app has all the functionality you were looking for when you first asked this question: it displays calendars for the upcoming months on your Apple Watch or your iPhone's Today View.