Hi GatorGirl95,
If you'd like to know more about muting alerts in the Messages app, this might help.
Are talking about what it shown in the section "Mute notifications for a conversation" in Change message notifications on iPhone? If you tap the bell with the line through it in the messages list, all you are doing is muting the alert. You would still see the messages if you were to open the conversation. That setting is only muting the alert; it isn't stopping messages from sending to your device.
Let the community know if you need further clarification.
We hope that helps with your question and take care!