Many Motorola Moto G users has been experiencing this issue however this bug is not limited to this phone model. Different experiences reported by users showing that the pop up message appears accidentally while other users reported this message appears upon receiving incoming call. Off course it is a very disturbing experience.
There are several approaches to address this issue. However, it is highly recommended to back up your contact before trying any of these approaches.
Possible fixes Moto G Bug: Unfortunately, Contacts has Stopped
1. Enter the Settings menu, choose the Apps section and then scroll over to the All tab. On this tab, look for Contacts and Contacts Storage, then tap Clear cache and Clear data button.
2. It is highly possible that another apps trying to sync or backup contacts causing this bug to appear. It is reported that signing out Google+ app as well as disable the app can address this issue with high success rate. Other apps that can be related with this issue are including Facebook and any other apps with access to your contacts data.
3. Another worthy shot is cache partition clearing. Clearing cache partition is more like removing temporary files. This approach can sometimes improves handled performance. It would be much useful to do after you updated the Android version.
For example you have the Lollipop version running on your handled, you need to get the phone turned off. Press and hold both Volume down and Power buttons together for a while until boot options menu appears on the screen. Find the Recovery using Volume down and select it using Volume up button.
When you see the fallen Android on screen, hold both Power and Volume up button at the same time and it will lead you to android recovery menu. Find wipe cache partition using Volume down and select it with Power button. Once it is completed, select Reboot system now using Power button.