This can be tricky like you said as most sites frown upon reviews that are listed this way ad this can have negative consequences. However, here are few things that also come to mind when encouraging positive reviews for your apartment complex:

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1. Try to personally follow-up after tenants have had their maintenance requests completed. If you do this via email you can easily link your preferred sites for reviews (Yelp, Google My Business, Website) and ask them to please leave quick feedback if they are happy with their service. If you guys respond quickly and the solution is effective, this will make it easier for them to just submit a review after a positive experience with management.
2. This also goes without saying, if you have email blasts — even for routine updates on water-shut offs or changes in office hours, link your sites for review in the social section so they can easily access.
3. Respond to the negative reviews — show future tenants that even if there was an issue your management team owned the issue and did their best to reach out and resolve the issue.
4. If you have social media, definitely utilize stories or content to encourage followers to leave reviews. You can even create a an IG Highlight to save positive reviews right at the top of your profile so they’re the first thing people see.
5. Create small flyers or postcards that are easy to hand out in the office/ around complex after a positive interaction with tenants. Simple enough and can be effective at helping tenants remember that they wanted to leave one when they see it inside their apartment.
6. Provide incentives for property managers and other employees to get in the habit of asking for reviews while they interact with tenants/ potential tenants can also help build up a more positive reputation online.