Can I appear to for Judiciary exam if I have backlogs in LLB, and if I clear this backlog, will I be eligible for judiciary exam?

The eligibility criteria for Judiciary exams may vary depending on the state or the conducting authority. However, most of the states require that the candidate should possess a law degree from a recognised university with a minimum percentage of marks, typically 50% for the general category and 45%

Yes definitely you can. After clearing your backlog you can definitely opt for judicary exams.