Is it possible to specify a custom date format instead of using
  ToString(“g”)?
Here is one possible answer. It converts to time in following format: 
day/month/year hours:minutes
 var dateTime = DateTime.Now.ToString("dd/MM/yyyy HH:mm");
Is there a function on the datetime object that outputs in this format?
There are similar methods like ToShortDateString() but it is still not same format like you described.
Do I have to create a custom extension method or go down some other
  path?
Based on my personal opinion you can create extension method with more descriptive name, but I rather like to declare some constant variable which hold value about your specific format and populate that variable from e.g. config file.
var dateTime = DateTime.Now.ToString(_someMoreDescriptiveDateTimeFormat);
Maybe good idea for extension method method can be like this:
        var dateTimeString = DateTime.Now.ToFormattedString();
And here is implementation of method which is documented with explanatory data. It can also be helpfull during review.
        /// <summary>
        /// Convert DateTime object into specially formated string.
        /// </summary>
        /// <param name="dateTime">DateTime which will be converted.  </param>
        /// <returns>Custom date and time string.</returns> 
        public static string ToFormattedString(this DateTime dateTime)
        {
            return dateTime.ToString("dd/MM/yyyy HH:mm");
        }