Is it possible to integrate Amazon QuickSight dashboard graphs to a web application?

• I don't think so. Even if you want to share the dashboard to someone, you need to create a user in QuickSight. Any more than 1 user will be charged by AWS.
• The dashboard cannot be public and you need to login to view the dashboard. If it was public, you could have embedded it in your webpage as an iframe. But you cannot.
• So, I think you are having limited options here, when it comes to QuickSight.
• You can always using D3 or Google Charts to display the data by exposing REST services for your data in MySQL.
• If you have a huge database, you may want to consider indexing the data to Elasticsearch and perform queries on it.
• Check if Kibana + Elasticsearch works out of the box for you.

Good luck!

Update: Dec 28 2018

Amazon announced in Nov 2018, that Amazon QuickSight dashboards can now be embedded in applications. Read more here at this AWS QuickSight Update.

AWS has enabled the embedding of the Dashboards into web apps. The feature was released on 27th Nov 2018. Here are a few helpful links:
1. https://aws.amazon.com/blogs/big-data/embed-interactive-dashboards-in-your-application-with-amazon-quicksight/
2. https://docs.aws.amazon.com/quicksight/latest/user/embedded-dashboards-setup.html