How can I export drawing from Notes to an image with background on an iPad? When I save drawing from the Notes app to Photos, the saved image has a transparent background instead of the background I see in the Notes app.

You can see the transparent background if you open the image in Google Photos or share the picture to any other app, e.g. opening that in the Preview app on macOS.

It is annoying when I share the image to some people and they couldn't see it clearly because of the transparent background.

Is there any way to export the drawing with the background (grids and lines) I've selected for the note?
A third, more crude method, which you can use if a certain part of your doodle requires background lines/grids is taking  screenshot of the doodle.

Step 1;
Expand the note using the button in the top right of the screen.

Step 2;
Hide the panel showing pens and colors using the “x” at the bottom right of the panel.

Step 3;
Take a screenshot by pressing the home and power buttons at the same time.

Step 4; Crop the screenshot to show only the desired result.

The end result is something like this;