The most "user friendly" solution is to establish a mostly stable interface between A and B and keep B as backwards compatible as possible over longer periods of time with any new version deployed to the web.
That gives users of A the opportunity to work with the version they are happy with as long as they can. It gives them also the opportunity to switch back to an older version of A when it turns out the new version has some annoying bugs your team is going to fix only next quarter.
Of course, you may be inclined not to support a five years old version of A with B when only 1% of your user base still uses that old version. Hence you should keep track which version of A really is in use, and yes, A should always send its version number to B. This gives you the opportunity to check if you really need to support deprecated features in B.
It will also give you the opportunity to implement a smooth transition process to a newer version. For example, A could show a warning like *"You are still using Version 1.0, which will not be supported by service B next quarter any more. Please ask your administrator to install version 5.0 or newer."
However, I would only force users to upgrade to a newer version when they really must and you definitely cannot affort to stay backwards compatible any more. I cannot tell you what a reasonable time interval for such forced upgrades is in your case, but when the installation of an upgrade involves some manual effort of an external IT department in the user's companies, better plan for years, not for quarters.