When I copy and paste a page…
Not clear enough. A web page that is displaying card suits?
If so, when you paste into another app, it is almost always going to paste as the default font for that app. Meaning, it is pasting the correct Unicode information for that glyph's font position, but not in the correct font. Then you get whatever is in the default font for those positions (if there's anything in that font at all for those Unicode locations).
Assuming the web page you're copying them from is using Symbol to display card suits as a font, you should be able to paste, then highlight the characters (no matter what they look like at the moment), and then choose Symbol from the font list for that app.
Or, skip the web page entirely and enter the glyphs on your own. For Symbol it's:
Hearts - Option+G
Clubs - Option+6
Diamonds - Shift+Option+U
Spades - Option+9
However, those are older Classic (pre-Unicode) keystrokes. I don't think they work in newer, Unicode aware apps.