Over overloaded method
As for me, this code is fine. Maybe there's some Reflection dark magic available but this one will be anyway faster.
The only thing I can suggest here is optimizing out an async State Machine because there's no code after await and the only await in the method can be executed.
Also note that you lookup through the Dictionary twice, first time when you check key with .ContainsKey, second time when you get the value with the same key. It can be done with single lookup using Dictionary.TryGetValue() method.
Also I suggest to follow async naming policy suggested by Microsoft, and rename the method to InvokeNotificationAsync.
private Task InvokeNotificationAsync(string userName, string methodName, params object[] parameters)
{
    if (connections.TryGetValue(userName, out var connection))
    {
        var proxy = Clients.Clients(connection);
        switch (parameters.Length)
        {
            case 0:
                return proxy.SendAsync(methodName);
                break;

            case 1:
                return proxy.SendAsync(methodName, parameters[0]);
                break;

            case 2:
                return proxy.SendAsync(methodName, parameters[0], parameters[1]);
                break;

            default:
                throw new HubException($&quot;No method was defined with {parameters.Length} parameters in Hub&quot;);
        }
    }
}

Also it can look a bit better with C# 8.0 syntax
private Task InvokeNotificationAsync(string userName, string methodName, params object[] parameters)
{
    if (connections.TryGetValue(userName, out var connection))
    {
        var proxy = Clients.Clients(connection);
        return parameters.Length switch
        {
            0 =&gt; proxy.SendAsync(methodName),
            1 =&gt; proxy.SendAsync(methodName, parameters[0]),
            2 =&gt; proxy.SendAsync(methodName, parameters[0], parameters[1]),
            _ =&gt; throw new HubException($&quot;No method was defined with {parameters.Length} parameters in Hub&quot;)
        };
    }
}