Is it possible to create a custom operator in Java?

No, Java is not extensible in this way. You can't add operators, and you can't even further overload built-in operators like + - even standard library classes like BigInteger have to use methods such as add() rather than operators such as +.

Scala (another static JVM language) gets around this by using method calls rather than built-in operators, and allowing any characters in method names, so you can define new methods that appear to be operators, i.e.

a + 1

is syntactic sugar for:

a.+(1)

Java doesn't allow for this.

However, if you want to achieve this sort of syntax whilst being able to run your code on a JVM (and with other Java code), you could look at Groovy, which has operator overloading (and with which you could also use DSLs for short syntax which would have similar effects to using custom operators).

Note that defining custom operators (not just overloading) is a big deal in any language, since you would have to be able to alter the lexer and grammar somehow.