In Configure Services in Startup:

services.AddAzureClients(builder =>
{
  // Use the environment credential by default
  builder.UseCredential(new DefaultAzureCredential()); 
  builder.AddQueueServiceClient(Configuration.GetSection("StorageConnectionString"))
    .ConfigureOptions(c => c.MessageEncoding = Azure.Storage.Queues.QueueMessageEncoding.Base64);
});

The implementation in the class is:

    public class AzureQueueService : IQueueService
    {
        private readonly QueueServiceClient _queueServiceClient;

        public AzureQueueService(QueueServiceClient queueServiceClient)
        {
            this._queueServiceClient = queueServiceClient;
        }

        public void SendMessageToQueue(MyMessage message)
        {
            this._queueServiceClient.GetQueueClient("my-queue-name")
                .SendMessage(JsonSerializer.Serialize(message, new JsonSerializerOptions { PropertyNamingPolicy = JsonNamingPolicy.CamelCase }));
        }
    }

an interface can be created with function/property which will return queue client and then implement the interface to inject in DI, hope below code will help.

public interface IQueueService
{
    QueueClient GetQueueClient();
}


    public class AzureQueueService : IQueueService
    {
        string azureStorageAccountConnectionString;
        string queueName;
    QueueClient queueClientInstance;

        public AzureQueueService(IConfiguration configuration)
        {

            azureStorageAccountConnectionString = configuration[Constants.KEY_AZURE_STORAGE_ACCOUNT_CONNECTION_STRING];
            queueName = configuration[Constants.KEY_QUEUE_NAME];
            queueClientInstance = new QueueClient(azureStorageAccountConnectionString, queueName);
    }

    public QueueClient GetQueueClient()
    {
      return queueClientInstance ;
    }
  }

Now you can inject AzureQueueService in dependency injection