Sure you can. Check these links out:
Consider this as an example card:
https://trello.com/c/NIRpzVDM
You can link to a card as a photo using the following:
<img src="https://trello.com/c/NIRpzVDM.png" />

You could embed an iframe as follows:
<iframe src="https://trello.com/c/NIRpzVDM.html"></iframe>
You can also do a javascript that writes the card into your page:
<script src="https://trello.com/c/NIRpzVDM.js"></script>
which writes out some html using document.write("<html content>")
Another way is to request a card's JSON
https://trello.com/c/NIRpzVDM.json
All of these are available using a board as well...
https://trello.com/b/nC8QJJoZ.png
https://trello.com/b/nC8QJJoZ.html
https://trello.com/b/nC8QJJoZ.js
https://trello.com/b/nC8QJJoZ.json