No, you may not trade game pieces.
The rules have this on trading:
Afterwards you may trade freely (using either or both types of trades below) to gain needed Resource Cards:
a) Domestic Trade
On your turn, you can trade Resource Cards with any of the other players. You can announce which resources you need and what you are willing to trade for them. The other players can also make their own proposals and counter offers.
Important: Players may only trade with the player whose turn it is. The other players may not trade among themselves.
b) Maritime Trade
You can also trade without the other players! During your turn, you can always trade at 4:1 by putting 4 identical Resource Cards back in their stack and taking any 1 Resource Card of your choice for it. If you have a settlement or city on a harbor, you can trade with the bank more favorably: at either a 3:1 ratio or in special harbors (trading the resource type shown) at 2:1.
Important: The 4:1 trade is always possible, even if you do not have a settlement on a harbor.
It is allowed to trade the resources for a road for some other resources, with this restriction: You are not allowed to trade the same goods. This means you cannot exchange brick for brick, or in this case 4 bricks for a brick and a wood. I believe handing over entire roads, settlements and/or cities worth of resources is tactically a dubious move for your opponents under most circumstances, but it is allowed.
You may not trade like resources (e.g., 2 wool for 1 wool).