Functional programming has two primary qualities that define it - emphasis on immutable data and first-class functions. Both are - with some effort and within reason - achievable in Java.
You can certainly have immutable objects. Have the object state explicitly set through the constructor, do not expose setters and apply the
final keyword to the fields. Whenever you would have a method modify the object's internal state, have it return a new instance with the updated state instead. An approach like this was presented in Real-World Functional Programming, a book containing examples of functional code in F# and corresponding code in C#. While C# is better suited for functional programming than Java, a similar approach could still be applied to Java.
When it comes to first-class functions, things get a bit more tricky. At the moment Java has a rather clumsy and verbose mechanism of emulating functions using anonymous nested classes, with a limited support for closures. While it's enough to write at least some pieces of code in a functional style, it would be a lackluster experience compared to using a proper functional language. There are Java libraries available, most notably guava, that introduce some of the functional programming concepts into Java, like immutable collections and some functional idioms. That would probably make the experience less gruesome - yet even the official documentation for guava states that functional idioms should only be used sparingly, as they often lead to code that is more verbose, less clear and less efficient than the corresponding imperative version.
The situation for Java is reportedly going to get better with the release of SE 8, set to include proper closures and lambda expressions, deferred from SE 7. Meanwhile you can try one of JVM's several functional languages and interop with Java when needed.