There is no Slovene or Slavic gene as such. There are certain markers such as Y-chromosome markers commonly found in different populations. In other words a certain genetic marker can be found among any European ethnicity. It’s just some genetic markers are more common to Slavic populations than to Romance populations. European populations also show more genetic affinity to some populations than other.
For example Slovenian population shows more genetic affinity to Croatian and Slovak populations than to French or German populations.