Only if you neuter monarchs from ever doing anything.
The constitutional monarchies of places like the UK and Australia demonstrate that they are incompatible, because the liberal democracy has to effectively remove the monarch almost utterly, in order to even make a liberal democracy. Our monarchs don't -arch, the fact that they haven't been entirely eliminated is a testament to institutional inertia, not anything ideologically democratic about monarchy.
I imagine Traitorfish, certainly most people,
would define the British monarchy as a monarchy. If the question is intended to exclude constitutional monarchies then it is trivial; of course Monarchies in which an unelected monarch rules is not compatible with liberal democracy. It is not compatible with democracy.
With that in mind, I would question your assertion that the perseverance of the monarchy is purely an issue 'institutional inertia'. At least in the UK case (I can speak less authoritatively about the Australian case) the monarch performs some vital constitutional functions. The monarch enhances the political system, rather than degrades it.
As has been mentioned, first and foremost the monarch constitutes a viable figure for ceremonial head of state. The role of the head of state is to represent the entirety of the polity and the monarch is uniquely suited to this role. That is because the monarch is strictly non-partisan. Any other mechanism for choosing a head of state would require make the position divisive and partisan (the US presidency, for example). The role of the monarchy in addressing the nation with no allegiance but that to the nation is one that is impossible to replicate within the modern party system. Any alternative candidate would require the backing of some political party and no such candidate could act as the impartial figurehead instantiated in the monarchy. This is all the more important with the spread of devolution.
Moreover an elected head of state (in terms of a presidency) would undermine the (very successful) Westminster model of democracy. This relies on the sovereignty of parliament and the consensus that parliament is the only body with the legitimacy required to govern. An elected head of state could clearly claim legitimacy from national constituency and could leverage this legitimacy into political influence. This is not a good thing.
Finally, the international benefits of the monarchy are often ignored. The monarchy is what unites the commonwealth and thus creates and excellent network of diplomatic links. For Britain, the monarch constitutes and excellent international ambassador which greatly helps the country maintain close relationships with its former empire. An elected head of state would make this whole system unstable.
I am not claiming that the benefits of them monarchy are huge nor that republicanism would entail societal collapse. But I am claiming that the monarchy is a useful institution with quite tangible benefits, at least for the UK.
As long as all the kings and lords shut up and get the hell out of the way, I suppose it could work. But I can't see any way that it wouldn't work better if they were gone altogether.
See above: a case in which constitutional monarchy works better than a republic.