As @Łukasz Kwieciński said, you'll be getting an 'await' outside async function error. To check if the code works and let be the screen_name and description attributes, I made a class, assuming your whostartedfollowing and whoisbeingfollowed are of a class that resembles user:
class user:
    def __init__(self):
        self.screen_name = "screen name"
        self.description = "description"
        self.followers_count = 3
Coming to the actual part of the question: sending a message without awaiting for a command, you can send the message anytime you want. For example, you can send the message when the bot starts using its on_ready() method, something like this:
@client.event
async def on_ready(): #When the bot starts
    print(f"The bot is online and is logged in as {client.user}")
    #Do some scraping here and get the actual whostartedfollowing and whoisbeingfollowed (from Twitter).
    whostartedfollowing:user = user()
    whoisbeingfollowed:user = user()
    await sendDiscordMessage(whoisbeingfollowed, whostartedfollowing)
    print("sent the message")
in which sendDiscordMessage() is the function in your code, but a bit modified to reduce errors:
async def sendDiscordMessage(friendUser, eachUserToScrape):
    friendUserScreenName = friendUser.screen_name
    friendUserDescription = friendUser.description
    friendUserFollowersCount = str(friendUser.followers_count)
    
    discordMessage = "\n\n=======================================================\n\n" + "** User @" + eachUserToScrape.screen_name + " started following : @" + friendUserScreenName + " **\n\nLink profile : https://twitter.com/" + friendUserScreenName + "\n\nSubscribers count : "+ friendUserFollowersCount + "\n\nDescription : " + friendUserDescription
                                                                                                        #Since screen_name is a string, I just used it instead of the object, so it can be joined with the rest as a string.
    channel = client.get_channel(...) #Get the channel with its ID (an `int`), for example a channel with 
    print(channel) #To check if the channel is None or not.
    print(discordMessage) #and not "print(discordMessage + type(discordMessage))", or you can use "print(discordMessage + str(type(discordMessage)))"
    await channel.send(discordMessage)
You can do the same thing without awaiting, by not making your sendDiscordMessage(friendUser, eachUserToScrape) function async (letting it be sendDiscordMessage(friendUser, eachUserToScrape) and not adding async to make it async def sendDiscordMessage(friendUser, eachUserToScrape)), if you don't want to depend on the result of the awaited coroutine. But, you'll get a RuntimeWarning: coroutine 'Messageable.send' was never awaited.
If you have to wait for the result of the awaited coroutine, where if the awaited coroutine crashes, then the rest of the code won't work either, or where the order of execution is important, then use await and change def sendDiscordMessage(friendUser, eachUserToScrape) to:
async def sendDiscordMessage(friendUser, eachUserToScrape):
Also, you're using discord.Client and not discord.ext.commands.Bot. The latter is just an extension of the former, with more features. You may want to switch to commands.Bot.
Here's the full code:
from discord.ext import commands
client = commands.Bot(command_prefix="")
class user:
    def __init__(self):
        self.screen_name = "screen name"
        self.description = "description"
        self.followers_count = 3
@client.event
async def on_ready(): #When the bot starts
    print(f"Bot online and logged in as {client.user}")
    #Do some scraping here and get the actual whostartedfollowing and whoisbeingfollowed (from Twitter).
    whostartedfollowing:user = user()
    whoisbeingfollowed:user = user()
    await sendDiscordMessage(whoisbeingfollowed, whostartedfollowing)
    print("sent the message")
async def sendDiscordMessage(friendUser, eachUserToScrape):
    friendUserScreenName = friendUser.screen_name
    friendUserDescription = friendUser.description
    friendUserFollowersCount = str(friendUser.followers_count)
    
    discordMessage = "\n\n=======================================================\n\n" + "** User @" + eachUserToScrape.screen_name + " started following : @" + friendUserScreenName + " **\n\nLink profile : https://twitter.com/" + friendUserScreenName + "\n\nSubscribers count : "+ friendUserFollowersCount + "\n\nDescription : " + friendUserDescription
                                                                                                        #Since screen_name is a string, I just used it instead of the object, so it can be joined with the rest as a string.
    channel = client.get_channel(...)
    print(channel)
    print(discordMessage) #and not "print(discordMessage + type(discordMessage))", or you can use "print(discordMessage + str(type(discordMessage)))"
    await channel.send(discordMessage)
client.run("token")
Here are some uses of async-await.