If you are using an older theme, it's probably using wp_list_pages to generate your nav menus. Two points:
1) It's actually the 'title' attribute, not the 'alt' attribute, that is causing the tooltip to pop up.
2) Filtering the output of wp_list_pages is quite difficult -- I know because I wrote a plugin that does it (but for a different reason).
$('#nav a').attr('title', '');
ekdor and PGrizz,
Something similar should work for you, assuming jQuery is loaded. You'll have to modify the CSS selector in the code to match your theme.