A really nice, but far from typically Dutch city!
The better database to use is
Waarneming.nl (just set it to English).
Or set it to the province level at
Vogelarchief Limburg (you can also search per site/municipality etc.)
The Sint-Pieterberg southwest of Maastricht has breeding Eagle Owl (it's obscured, but the viewpoint is called "Kiekoet Oehoevallei" – google maps will guide you).
You should be able to find all woodpeckers in nearby forests (waarneming.nl will be helpful). The forests east of Epen (Vijlenerbos) have the woodpeckers as well as Eurasian Treecreeper, which is far less common than Short-toed.
For general birding, the Eijsder Beemden and Itteren are probably the place to be. These areas attract migrants along the river Meuse ("Maas" in Dutch) and could hold some surprises.
The fields north of Groot-Welsden are managed for Field Hamster (Sibbe, Hamsterreservaat) and can be good for Hen Harrier and some of the last Dutch Corn Buntings.
Non-bird distractions: April is a good time for forest flora – Savelsbos and Bunderbos near Maastricht are very nice, if not particularly exciting ornithologically. It's a bit early for the chalk grassland flora or butterflies.