I've never used Migration Assistant, so can't say if it would carry Java 6 over, but probably so.
In the root Library folder (not the one with the System folder or your user account), if you can navigate to:
/Library/Java/JavaVirtualMachines/1.6.0.jdk
Then Java 6 is installed. You can double prove it by opening Terminal in the /Applications/Utilities/ folder and typing:
java -version
It should return something like this:
java version "1.6.0_65"
Java(TM) SE Runtime Environment (build 1.6.0_65-b14-468-11M4828a)
Java HotSpot(TM) 64-Bit Server VM (build 20.65-b04-468, mixed mode)
The 6 means version 6, and the 65 is the release number (version 65 of Java 6). It's unknown why Oracle puts the preceding 1 or trailing 0 in the version numbers for Java.