No biodiversity index is an end in itself, it is only a tool in telling a larger story. Let's say that they tell the opposite story, one index tells that you have a high diverse community and the other that you have low diversity community, what would each reveal about your community? There might be a feature of one of the indices that is not applicable to your situation, so it's possible that only one is correctly used, and the other isn't currently applicable. There is also a possibility that they take slightly different features into consideration while calculating the index, such as like abundance, evenness, distinctiveness etc, leading to different conclusions. If neither of these two examples are applicable to your situation, and both are used correctly, then you have to figure out which index is most applicable to your situation, and motivate your choice. By including both, if they point in different directions, you'll have a difficult time to weave a convincing narrative.

I think that this paper - Brown et al. 2006. ( Species diversity in the Florida Everglades, USA: A systems approach to calculating biodiversity) - has a lot of helpful background about the approach to calculating biodiversity and why different metrics matter and perhaps provides a useful theoretical springboard for your analysis, hope it's helpful!

It's common to report more than one index. Diversity indices aren't like statistics where you have parametric and non-parametric methods, and only one is appropriate based on the data. They are attempts to describe different aspect of diversity, such as richness, evenness, rareness, etc. Ultimately, it's up to the researcher to interpret what the index is saying because it's just a number from a formula that is meaningless without context and a description. The tool needs to fit the question and what you're seeking to find.