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First, is this part correct?

} else if (str.length() == 1 && str.charAt(0) == 'a') {
    return str;
} else {
    return str;
}

This will return a string of length 1, unchanged, regardless of the character. What should happen if you pass "z"? Since the first character is not 'a', should it be removed?

I suspect it was supposed to be something like this:

} else if (str.length() == 1 && str.charAt(0) != 'a') {
    return "";
} else {
    return str;
}

The rest of this answer assumes it should remove the character

You could use a StringBuilder and delete the characters as necessary:

public String deFront(String str) {
    StringBuilder sb = new StringBuilder(str);
    if (sb.length() > 1 && sb.charAt(1) != 'b') {
        sb.deleteCharAt(1);
    }
    if (sb.length() > 0 && sb.charAt(0) != 'a') {
        sb.deleteCharAt(0);
    }
    return sb.toString();
}

Or you could use a regular expression search & replace:

public String deFront(String str) {
    str = str.replaceFirst("^(.)[^b]", "$1");
    str = str.replaceFirst("^[^a]", "");
    return str;
}

Note: The order of deletion is important in both of these solutions. The b is deleted first to prevent the deletion of a from shifting the b by one position.