I have a dragonborn character that originally started in skyrim. In Skyrim there is a powerful conjure spell which is Conjure Ice Dragon which I have made a staple as apart of her spellbook. Dragons are labelled under beasts. Is it possibly to do this?

Dragons aren’t beasts, they’re dragons. And I’m not sure what exactly you’re asking. Do you mean “is it possible to do in D&D Beyond’s homebrew spell editor”?

First, the spell you mention does not exist in Skyrim without mods. (Unless you're counting Dragon Shouts, which aren't spells).

As Saga mentioned in D&D dragons have their own type and are not beasts.

Conjure Animals is limited to beasts anyway.

Without homebrewing no spell exists to conjure dragons. The closest is 8th level Illusory Dragon which creates the illusion of an Adult Dragon of choice, but is not the real thing. Alternatively Tru Polymorph or Shapechange spells can let you or another become a dragon (even permanently for True Poly).

Can homebrew be done? Yes. There's not much you cannot make as homebrew, be it with the D&D Beyonds or without. You could make a 1st level spell that can create solar systems if you wanted. The true deciding factor on whether you get to use it in a game is the DM. I cannot imagine many DMs allowing you to conjure dragons with any spell less than 9th level, or at all. Dragons are powerful. Shapechange / True Poly balances by taking out your ally and replacing them with the dragon. Illusory Dragon is balanced because it's an illusion and on realising it the enemy can halve any damage.

I'd recommnd sticking with True Poly/Shapechange/Illusory Dragon.