Yes, it is possible!
webView.setBackgroundColor(0x00FFFFFF);
However, if in your AndroidManifest.xml contains:
android:hardwareAccelerated="true"
webView's background, for some reason, does not change to transparency.
(It took me while to figure this out!)
This is how to disable hardware acceleration on the view that contains webview:
view.setLayerType(View.LAYER_TYPE_SOFTWARE, null);