Part of the problem here is that you can't open a conversation window without knowing who you want to start a conversation with, if that makes sense.
From a web-page, there are various different things you could do to start an IM with someone, for instance, Lync has a shell extension so you could start the process im:<sip:[email protected]>
and it will open a new conversation window with that user. However, there's no way to supply a default message text.
I'm assuming that, given you're assuming the Lync Client is available, this is for an internal environment you have some control over? If that's the case, if you can find out the recipient's address ahead of time (see first sentence) then you could write a really small console application which took the SIP address as a parameter, and started a new conversation sending the default message as well. You could then call this from your webpage. You'd have to deal with deployment of your console app etc, but it would get you where you need to be.
This MSDN article is a walk-through of starting a conversation, though you won't need all of it as you won't need to deal with receiving messages.