It is possible and you do not need a plugin for it (Assuming you are using QGIS3)
Zoom to an area covering where you want the points
Then, Go to Vector -> Research tools -> Regular points
Set input extent to Use Canvas extent and set point spacing to 1
You will probably get more points than you want / need, so for next step select the polygon you want to have the points in,
Go to Vector -> Geoprocessing tools -> Clip
Select the newly created layer with the points as the input layer, Select the layer with your polygon as the Overlay layer, check Selected features only and "Run"
Just two details: You probably want to have your project in a meter based projection before doing this and if you do not save / make permanent your newly created layer, it will be gone when you close QGIS.
If you want to be able to display points in one or more polygons in your layer, first make the points over your entire layer, then for the second point use vector -> geoprocessing -> intersection using your newly created points and your polygon layer and you will get a new point layer with the attributes copied from the polygons.